3.58.69 \(\int \frac {-4 x+(-4+e^x (1-x)) \log (x^4)+((-4+e^x-x) \log (x^4)+x \log (x^4) \log (\log (x^4))) \log (\frac {x}{-8+2 e^x-2 x+2 x \log (\log (x^4))})}{(-4+e^x-x) \log (x^4)+x \log (x^4) \log (\log (x^4))} \, dx\) [5769]

Optimal. Leaf size=25 \[ x \log \left (\frac {x}{2 \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )}\right ) \]

[Out]

x*ln(x/(2*x*ln(ln(x^4))+2*exp(x)-2*x-8))

________________________________________________________________________________________

Rubi [A]
time = 2.58, antiderivative size = 26, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 4, integrand size = 97, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {6873, 6874, 6820, 2629} \begin {gather*} x \log \left (-\frac {x}{2 \left (x \left (-\log \left (\log \left (x^4\right )\right )\right )+x-e^x+4\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x + (-4 + E^x*(1 - x))*Log[x^4] + ((-4 + E^x - x)*Log[x^4] + x*Log[x^4]*Log[Log[x^4]])*Log[x/(-8 + 2*E
^x - 2*x + 2*x*Log[Log[x^4]])])/((-4 + E^x - x)*Log[x^4] + x*Log[x^4]*Log[Log[x^4]]),x]

[Out]

x*Log[-1/2*x/(4 - E^x + x - x*Log[Log[x^4]])]

Rule 2629

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*Simplify[D[u, x]/u], x], x] /; ProductQ[
u]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 x-\left (-4+e^x (1-x)\right ) \log \left (x^4\right )-\left (\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )\right ) \log \left (\frac {x}{-8+2 e^x-2 x+2 x \log \left (\log \left (x^4\right )\right )}\right )}{\log \left (x^4\right ) \left (4-e^x+x-x \log \left (\log \left (x^4\right )\right )\right )} \, dx\\ &=\int \left (1-x+\frac {x \left (-4-3 \log \left (x^4\right )-x \log \left (x^4\right )-\log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )\right )}{\log \left (x^4\right ) \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )}+\log \left (\frac {x}{2 \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )}\right )\right ) \, dx\\ &=x-\frac {x^2}{2}+\int \frac {x \left (-4-3 \log \left (x^4\right )-x \log \left (x^4\right )-\log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )\right )}{\log \left (x^4\right ) \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )} \, dx+\int \log \left (\frac {x}{2 \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )}\right ) \, dx\\ &=x-\frac {x^2}{2}+x \log \left (-\frac {x}{2 \left (4-e^x+x-x \log \left (\log \left (x^4\right )\right )\right )}\right )-\int \frac {-4 x+\left (-4-e^x (-1+x)\right ) \log \left (x^4\right )}{\log \left (x^4\right ) \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )} \, dx+\int \frac {x \left (4-\log \left (x^4\right ) \left (-3-x+(-1+x) \log \left (\log \left (x^4\right )\right )\right )\right )}{\log \left (x^4\right ) \left (4-e^x+x-x \log \left (\log \left (x^4\right )\right )\right )} \, dx\\ &=x-\frac {x^2}{2}+x \log \left (-\frac {x}{2 \left (4-e^x+x-x \log \left (\log \left (x^4\right )\right )\right )}\right )+\int \left (-\frac {3 x}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )}-\frac {x^2}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )}-\frac {4 x}{\log \left (x^4\right ) \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )}-\frac {x \log \left (\log \left (x^4\right )\right )}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )}+\frac {x^2 \log \left (\log \left (x^4\right )\right )}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )}\right ) \, dx-\int \left (1-x+\frac {x \left (-4-3 \log \left (x^4\right )-x \log \left (x^4\right )-\log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )\right )}{\log \left (x^4\right ) \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )}\right ) \, dx\\ &=x \log \left (-\frac {x}{2 \left (4-e^x+x-x \log \left (\log \left (x^4\right )\right )\right )}\right )-3 \int \frac {x}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )} \, dx-4 \int \frac {x}{\log \left (x^4\right ) \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )} \, dx-\int \frac {x^2}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )} \, dx-\int \frac {x \log \left (\log \left (x^4\right )\right )}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )} \, dx+\int \frac {x^2 \log \left (\log \left (x^4\right )\right )}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )} \, dx-\int \frac {x \left (-4-3 \log \left (x^4\right )-x \log \left (x^4\right )-\log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )\right )}{\log \left (x^4\right ) \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )} \, dx\\ &=x \log \left (-\frac {x}{2 \left (4-e^x+x-x \log \left (\log \left (x^4\right )\right )\right )}\right )-3 \int \frac {x}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )} \, dx-4 \int \frac {x}{\log \left (x^4\right ) \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )} \, dx-\int \frac {x^2}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )} \, dx-\int \frac {x \log \left (\log \left (x^4\right )\right )}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )} \, dx+\int \frac {x^2 \log \left (\log \left (x^4\right )\right )}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )} \, dx-\int \frac {x \left (4-\log \left (x^4\right ) \left (-3-x+(-1+x) \log \left (\log \left (x^4\right )\right )\right )\right )}{\log \left (x^4\right ) \left (4-e^x+x-x \log \left (\log \left (x^4\right )\right )\right )} \, dx\\ &=x \log \left (-\frac {x}{2 \left (4-e^x+x-x \log \left (\log \left (x^4\right )\right )\right )}\right )-3 \int \frac {x}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )} \, dx-4 \int \frac {x}{\log \left (x^4\right ) \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )} \, dx-\int \frac {x^2}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )} \, dx-\int \frac {x \log \left (\log \left (x^4\right )\right )}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )} \, dx+\int \frac {x^2 \log \left (\log \left (x^4\right )\right )}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )} \, dx-\int \left (-\frac {3 x}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )}-\frac {x^2}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )}-\frac {4 x}{\log \left (x^4\right ) \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )}-\frac {x \log \left (\log \left (x^4\right )\right )}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )}+\frac {x^2 \log \left (\log \left (x^4\right )\right )}{-4+e^x-x+x \log \left (\log \left (x^4\right )\right )}\right ) \, dx\\ &=x \log \left (-\frac {x}{2 \left (4-e^x+x-x \log \left (\log \left (x^4\right )\right )\right )}\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 25, normalized size = 1.00 \begin {gather*} x \log \left (\frac {x}{2 \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x + (-4 + E^x*(1 - x))*Log[x^4] + ((-4 + E^x - x)*Log[x^4] + x*Log[x^4]*Log[Log[x^4]])*Log[x/(-8
+ 2*E^x - 2*x + 2*x*Log[Log[x^4]])])/((-4 + E^x - x)*Log[x^4] + x*Log[x^4]*Log[Log[x^4]]),x]

[Out]

x*Log[x/(2*(-4 + E^x - x + x*Log[Log[x^4]]))]

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 5.64, size = 994, normalized size = 39.76

method	result	size
risch	\(\text {Expression too large to display}\)	\(994\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x*ln(x^4)*ln(ln(x^4))+(exp(x)-4-x)*ln(x^4))*ln(x/(2*x*ln(ln(x^4))+2*exp(x)-2*x-8))+((1-x)*exp(x)-4)*ln(x
^4)-4*x)/(x*ln(x^4)*ln(ln(x^4))+(exp(x)-4-x)*ln(x^4)),x,method=_RETURNVERBOSE)

[Out]

-x*ln(x*ln(4*ln(x)-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*csgn(I*x^3)*(-csgn(I*x^3)+csgn(I*x
^2))*(-csgn(I*x^3)+csgn(I*x))-1/2*I*Pi*csgn(I*x^4)*(-csgn(I*x^4)+csgn(I*x^3))*(-csgn(I*x^4)+csgn(I*x)))-x+exp(
x)-4)+x*ln(x)-1/2*I*Pi*x*csgn(I*x)*csgn(I/(x*ln(4*ln(x)-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*
Pi*csgn(I*x^3)*(-csgn(I*x^3)+csgn(I*x^2))*(-csgn(I*x^3)+csgn(I*x))-1/2*I*Pi*csgn(I*x^4)*(-csgn(I*x^4)+csgn(I*x
^3))*(-csgn(I*x^4)+csgn(I*x)))-x+exp(x)-4))*csgn(I*x/(x*ln(4*ln(x)-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x
))^2-1/2*I*Pi*csgn(I*x^3)*(-csgn(I*x^3)+csgn(I*x^2))*(-csgn(I*x^3)+csgn(I*x))-1/2*I*Pi*csgn(I*x^4)*(-csgn(I*x^
4)+csgn(I*x^3))*(-csgn(I*x^4)+csgn(I*x)))-x+exp(x)-4))+1/2*I*Pi*x*csgn(I*x)*csgn(I*x/(x*ln(4*ln(x)-1/2*I*Pi*cs
gn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*csgn(I*x^3)*(-csgn(I*x^3)+csgn(I*x^2))*(-csgn(I*x^3)+csgn(I*x))-
1/2*I*Pi*csgn(I*x^4)*(-csgn(I*x^4)+csgn(I*x^3))*(-csgn(I*x^4)+csgn(I*x)))-x+exp(x)-4))^2+1/2*I*Pi*x*csgn(I/(x*
ln(4*ln(x)-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*csgn(I*x^3)*(-csgn(I*x^3)+csgn(I*x^2))*(-c
sgn(I*x^3)+csgn(I*x))-1/2*I*Pi*csgn(I*x^4)*(-csgn(I*x^4)+csgn(I*x^3))*(-csgn(I*x^4)+csgn(I*x)))-x+exp(x)-4))*c
sgn(I*x/(x*ln(4*ln(x)-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*csgn(I*x^3)*(-csgn(I*x^3)+csgn(
I*x^2))*(-csgn(I*x^3)+csgn(I*x))-1/2*I*Pi*csgn(I*x^4)*(-csgn(I*x^4)+csgn(I*x^3))*(-csgn(I*x^4)+csgn(I*x)))-x+e
xp(x)-4))^2-1/2*I*Pi*x*csgn(I*x/(x*ln(4*ln(x)-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*csgn(I*
x^3)*(-csgn(I*x^3)+csgn(I*x^2))*(-csgn(I*x^3)+csgn(I*x))-1/2*I*Pi*csgn(I*x^4)*(-csgn(I*x^4)+csgn(I*x^3))*(-csg
n(I*x^4)+csgn(I*x)))-x+exp(x)-4))^3-x*ln(2)

________________________________________________________________________________________

Maxima [A]
time = 0.56, size = 31, normalized size = 1.24 \begin {gather*} -x \log \left (2\right ) - x \log \left (x {\left (2 \, \log \left (2\right ) - 1\right )} + x \log \left (\log \left (x\right )\right ) + e^{x} - 4\right ) + x \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*log(x^4)*log(log(x^4))+(exp(x)-4-x)*log(x^4))*log(x/(2*x*log(log(x^4))+2*exp(x)-2*x-8))+((1-x)*e
xp(x)-4)*log(x^4)-4*x)/(x*log(x^4)*log(log(x^4))+(exp(x)-4-x)*log(x^4)),x, algorithm="maxima")

[Out]

-x*log(2) - x*log(x*(2*log(2) - 1) + x*log(log(x)) + e^x - 4) + x*log(x)

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 22, normalized size = 0.88 \begin {gather*} x \log \left (\frac {x}{2 \, {\left (x \log \left (\log \left (x^{4}\right )\right ) - x + e^{x} - 4\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*log(x^4)*log(log(x^4))+(exp(x)-4-x)*log(x^4))*log(x/(2*x*log(log(x^4))+2*exp(x)-2*x-8))+((1-x)*e
xp(x)-4)*log(x^4)-4*x)/(x*log(x^4)*log(log(x^4))+(exp(x)-4-x)*log(x^4)),x, algorithm="fricas")

[Out]

x*log(1/2*x/(x*log(log(x^4)) - x + e^x - 4))

________________________________________________________________________________________

Sympy [A]
time = 8.63, size = 24, normalized size = 0.96 \begin {gather*} x \log {\left (\frac {x}{2 x \log {\left (\log {\left (x^{4} \right )} \right )} - 2 x + 2 e^{x} - 8} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*ln(x**4)*ln(ln(x**4))+(exp(x)-4-x)*ln(x**4))*ln(x/(2*x*ln(ln(x**4))+2*exp(x)-2*x-8))+((1-x)*exp(
x)-4)*ln(x**4)-4*x)/(x*ln(x**4)*ln(ln(x**4))+(exp(x)-4-x)*ln(x**4)),x)

[Out]

x*log(x/(2*x*log(log(x**4)) - 2*x + 2*exp(x) - 8))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*log(x^4)*log(log(x^4))+(exp(x)-4-x)*log(x^4))*log(x/(2*x*log(log(x^4))+2*exp(x)-2*x-8))+((1-x)*e
xp(x)-4)*log(x^4)-4*x)/(x*log(x^4)*log(log(x^4))+(exp(x)-4-x)*log(x^4)),x, algorithm="giac")

[Out]

integrate(-(((x - 1)*e^x + 4)*log(x^4) - (x*log(x^4)*log(log(x^4)) - (x - e^x + 4)*log(x^4))*log(1/2*x/(x*log(
log(x^4)) - x + e^x - 4)) + 4*x)/(x*log(x^4)*log(log(x^4)) - (x - e^x + 4)*log(x^4)), x)

________________________________________________________________________________________

Mupad [B]
time = 4.99, size = 25, normalized size = 1.00 \begin {gather*} x\,\ln \left (-\frac {x}{2\,x-2\,{\mathrm {e}}^x-2\,x\,\ln \left (\ln \left (x^4\right )\right )+8}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + log(x^4)*(exp(x)*(x - 1) + 4) + log(-x/(2*x - 2*exp(x) - 2*x*log(log(x^4)) + 8))*(log(x^4)*(x - exp
(x) + 4) - x*log(x^4)*log(log(x^4))))/(log(x^4)*(x - exp(x) + 4) - x*log(x^4)*log(log(x^4))),x)

[Out]

x*log(-x/(2*x - 2*exp(x) - 2*x*log(log(x^4)) + 8))

________________________________________________________________________________________